# Vector journeys

## Problem

*Vector Journeys printable sheet*

**Charlie likes to go for walks around a square park.**

Here is a diagram of the journey he took one day:

He started his journey by walking along the black vector $\pmatrix{3\cr 1}$

What vectors did he need to walk along to complete his journey?

Draw some other square parks that Charlie could walk around, and find the vectors that would describe his journey.

**Can you describe and explain any relationships between the vectors that determine Charlie's journey around any square park?**

Once you know the first vector of a journey, can you work out what the second, third and fourth vectors will be? Is there more than one possibility?

**Alison likes to walk across parks diagonally.**

One day, she walked along the blue vector $\pmatrix{2\cr 4}$:

- Can you describe any relationships between the vectors that determine Alison's and Charlie's journey, for any square park?

- Given the vector that describes Alison's journey, how can you work out the first stage of Charlie's journey?

- If all square parks have their vertices on points of a dotty grid, what can you say about the vectors that describe Alison's diagonal journey?

**Can you explain and justify your findings?**

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## Getting Started

The interactivity in Square Coordinates may help you to visualise tilted squares.

You can challenge yourself by playing Square It against the computer.

## Student Solutions

James from Wilson's School told us something he'd noticed about the second, third and fourth vectors forming a square park:

The vector journey goes: $\pmatrix{3\cr 1} \pmatrix{-1\cr 3} \pmatrix{-3\cr -1} \pmatrix{1\cr -3}$

After finding the first vector both top and bottom numbers should be positive, next vector the top number is made negative then next vector both numbers are negative and finally the last vector the bottom number is negative. The second and last vectors the left/right number switches with the up/down number.

Elliott, also from Wilson's, made some observations:

Charlie walks on vectors of $\pmatrix{3\cr 1} \pmatrix{-1\cr 3} \pmatrix{-3\cr -1} \pmatrix{1\cr -3}$

Another square he could walk would have vectors of $\pmatrix{5\cr 2} \pmatrix{-2\cr 5} \pmatrix{-5\cr -2} \pmatrix{2\cr -5}$

These vectors must only consist of four numbers: $x, y, -x$ and $-y$.

It can only be two numbers, and their negatives, so that all the sides of the square are equal in length.

After travelling along the first vector, you can then move left or right. From there, you must do the opposite of your first move, then the opposite of your second, to get back to your original position.

Josephine from The Urswick School added:

The vectors must add up to zero.

Niharika from Leicester High School for Girls sent us this solution. Well done to you all.

## Teachers' Resources

### Why do this problem?

This problem offers a simple context for exploring vectors that leads to some interesting generalisations that can be proved with some vector algebra.

Here is an article that describes some of the background thinking that informed the creation of this problem.

### Possible approach

*You may be interested in our collection Dotty Grids - an Opportunity for Exploration, which offers a variety of starting points** that can lead to geometric insights.*

This problem requires students to draw tilted squares reliably. This interactivity might be helpful to demonstrate to students what a tilted square looks like. Students could play Square It until they can reliably spot tilted squares on a dotty grid.

**A possible start which involves the minimum of teacher input** is to draw the vector $\pmatrix{3\cr 1}$ and say:

"Imagine we are drawing squares using vectors with whole numbers.

This vector could be the side of a square, or the diagonal of a square.

Find the vectors that describe the journeys around the squares that include this vector as either a side or a diagonal."

This leads on to the challenge "In a while, I am going to ask you to find the vectors that describe the journeys around squares that could be drawn using a **different** vector as either the side or the diagonal. The challenge will be to answer without doing any drawing."

**Alternatively, start** by showing the picture of Charlie's walk.

"If the black vector is $\pmatrix{3\cr 1}$ what are the other three vectors?"

Once everyone is confident with vector notation, ask students to draw a square park of their own on dotty paper, making sure the vertices are on lattice points, and to work out the vectors that would describe Charlie's journey.

On the board, draw a table to collect together some of the vector journeys the students have devised. After the first few, can they start predicting what the second, third and fourth vectors will be once they know the first vector of a journey? Is there more than one possibility?

Give students some time to work on their own or in pairs to test any conjectures they make.

"Could we have worked out the vectors if we'd been given a diagonal of the square instead of a side?"

Show Alison's diagonal walk, and ask students to consider this question with regard to the squares they drew earlier on. After a short while, the diagonal vectors could be added to the information already collected on the board.

Then set students the three questions from the problem:

- Can they describe any relationships between the vectors that determine Alison's and Charlie's journey, for any square park?
- Given the vector that describes Alison's journey, how can they work out the first stage of Charlie's journey?
- If all square parks have their vertices on points of a dotty grid, what can they say about the vectors that describe Alison's diagonal journey?

### Key questions

How can I use the first vector to work out the other three vectors which describe a journey around a square?

### Possible support

The interactivity in Square Coordinates helps students to visualise tilted squares.

### Possible extension

Vector Walk challenges students to explore relationships between vector algebra and geometry, and to consider the points that can be reached on a grid using a set of vectors.